## Abstract

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.

## 1. Introduction

Systems of quasi-linear first-order differential equations of the form
*ψ* obeying the system of linear hyperbolic equations
*A*_{ik} are defined by the system
*ψ*_{⋆} in the corresponding conservation laws are (uniquely) determined via integration of the compatible system
*ψ* and the flux *ψ*_{⋆} is of Combescure type [14]. In modern integrable system terminology, the densities *ψ* constitute eigenfunctions, whereas the characteristic speeds

Remarkably, Tsarev [7,8] has proved that, locally, all solutions of a semi-Hamiltonian system of the form (1.1) are given implicitly by the algebraic system
*ω*^{i}} denoting the general set of adjoint eigenfunctions obeying the linear system (1.3). This linearization technique has come to be known as the generalized hodograph method because, in the case *N*=1, the quantities *x* and *t* regarded as the unknowns of the system (1.6) obey the classical hodograph equations [15]
*N*. In summary, the properties of semi-Hamiltonian systems of the form (1.1) and their solutions are completely encoded in the classical surface theory of conjugate nets. This highlights the privileged nature of semi-Hamiltonian systems of hydrodynamic type.

The particular class of conjugate nets governed by the compatible hyperbolic equations
*ϵ*^{i}s constitute constants, plays a distinguished role in the theory of semi-Hamiltonian hydrodynamic-type systems (with the identification ** u**=

**). However, it is important to note that, in general, these special conjugate net equations are,**

*x**a priori*, unrelated to the conjugate net equations (1.2). In particular, the eigenfunction

*ϕ*does not necessarily play the role of a density. The linear equations (1.8) are known as Euler–Poisson–Darboux equations and have been the subject of extensive investigation in classical differential geometry [16]. Their importance in the one-phase Whitham equations for the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations has been observed in references [17,18,19] and, in the multi-phase case, in reference [20]. In fact, the explicit expressions for the characteristic speeds in the multi-phase Whitham equations derived in the pioneering paper [21] and also in reference [20] contain, as elementary building blocks, particular solutions of the Euler–Poisson–Darboux equations with parameters

*ϵ*

^{i}=1/2. Recently, it has been demonstrated [22] that the characteristic speeds for multi-phase Whitham equations may be obtained by means of iterated Darboux transformations generated by contour integrals of separable solutions of (extended) Euler–Poisson–Darboux systems. Euler–Poisson–Darboux systems for different values of

*ϵ*

^{i}also play a central role in the treatment of various dispersionless soliton equations and

*ϵ*-systems [23,24,25].

The connection between the Euler–Poisson–Darboux system and the characteristic speeds of multi-phase Whitham equations (and therefore the associated conjugate net system (1.2)) is provided by the observation that the hyperelliptic integrals
*b*_{κ}, *κ*=1,…,*g* are appropriately chosen cycles [20,21,26] and *k*≤*g*−1, may be regarded as superpositions of separable solutions of the Euler–Poisson–Darboux system (1.8) for *ϵ*^{i}=1/2. The methods recorded in [20,21,22] are then used in the (algebraic) construction of the characteristic speeds λ^{i}. For instance, in the case *g*=1, the elliptic integral (1.9) may be calculated to be essentially
*K* denotes the complete elliptic integral of the first kind. Each of the three coordinates *x*^{i} gives rise to a classical Levy transform [27]
*ϕ*^{O} and these coincide with the characteristic speeds for the one-phase Whitham equations [2]. The avatar (1.11) of these characteristic speeds may be found in references [19,28]. It is remarked in passing that the action of the Levy transformation on semi-Hamiltonian systems of hydrodynamic type has been discussed in detail in reference [29].

As indicated above, many systems of hydrodynamic type (1.1) admit dispersive counterparts which are integrable by means of the inverse spectral transform (IST) method [2,6]. One of the remarkable properties of IST integrable equations is that they admit integrable discretizations which reveal their fundamental properties [30]. Such discretizations are usually constructed via invariances of the integrable equations under Bäcklund, Darboux or similar discrete transformations [14]. This method simultaneously leads to a discretization of the underlying linear representation (Lax pair [31]).

Based on the standard discretization [32] of the conjugate net equations (1.2) and associated adjoint equations (1.5), we here propose a canonical integrability-preserving way of discretizing the theory outlined in the preceding. In particular, this is shown to lead to integrable discretizations of generalized hodograph equations, canonical cycles and associated hyperelliptic integrals and characteristic speeds of commuting flows of hydrodynamic type such as those corresponding to the multi-phase Whitham equations. It is noted that large classes of solutions of the standard discrete conjugate net equations may be obtained by means of, for instance, the ∂-bar-dressing method [33], Darboux-type transformations [34,35] or the algebro-geometric approach employed in reference [36]. Our approach exploits the existence of a canonical discretization of the classical Euler–Poisson–Darboux system (1.8) and associated separable solutions.

## 2. Generalized hodograph equations

We are concerned with commuting flows of diagonal systems of hydrodynamic type, that is, compatible systems of first-order equations of the type
*u*^{i} denote derivatives with respect to the independent variables *x* and *t*^{α}. It is emphasized that even though the above systems constitute the point of departure, many of the mathematical notions presented in this paper go beyond these systems and turn out to be of interest in their own right. It is known [8] that diagonal systems of hydrodynamic type commute if and only if the *N* sets of characteristic speeds *α* obey the same linear system
*A*_{ik} may be regarded as being defined by the equations for, say, *α*=1. Here, *ψ*_{α} defined (up to constants of integration) by
*ψ* is a solution of the linear hyperbolic equations
*A*_{ik} cannot be arbitrary as these are constrained by the compatibility conditions for the hyperbolic equations (2.4) or, equivalently, the first-order equations (2.2). In fact, the coefficients *A*_{ik} must be solutions of an integrable system of nonlinear partial differential equations known as the Darboux system. Indeed, in the context of the geometric theory of integrable systems (see, e.g. [14] and references therein), the function *ψ* constitutes an eigenfunction of the conjugate net equations (2.4) and the sets *ψ*_{α} are Combescure transforms of the eigenfunction *ψ* and, for reasons of symmetry, it is evident that each Combescure transform is a solution of another system of conjugate net equations with different coefficients.

### (a) The generalized hodograph method

In order to motivate the approach adopted in this paper, we here recall the generalized hodograph method developed by Tsarev in [8] for a single system of hydrodynamic-type equations
*A*_{ik} defined by (2.2), that is,
*ω*^{i}} is another set of adjoint eigenfunctions obeying the above linear system then any local solution ** u**(

*x*,

*t*) of the nonlinear system

*N*=1, (2.7) may be regarded as a linear system for

*x*and

*t*rather than a nonlinear system for

*u*

^{1}and

*u*

^{2}and differentiation of

*x*(

*u*

^{1},

*u*

^{2}) and

*t*(

*u*

^{1},

*u*

^{2}) leads to the classical hodograph system [15]

^{i}are regarded as known functions of the independent variables

*u*

^{k}. In the original context, this linear system is obtained from the nonlinear two-component system (2.5)

_{N=1}by merely interchanging dependent and independent variables, whereby the Jacobian determinant drops out.

### (b) Generalized hodograph equations

Even though the generalized hodograph method encapsulated in the algebraic system (2.7) is applicable for all *N*, an associated system of hodograph-type equations is not available for *N*>1, because the number of independent variables does not coincide with the number of dependent variables. However, because any flow which commutes with the hydrodynamic-type equations (2.5) does not impose any constraint on the space of solutions, it is natural to supplement (2.5) by *N*−1 commuting flows, leading to the larger system (2.1). Thus, if {*μ*^{i}} constitutes another set of adjoint eigenfunctions, then we may locally define a coordinate transformation
*N*=1.

It is now easy to see that Tsarev's generalized hodograph method is still valid in this more general setting so that, locally, the general solution of the hydrodynamic-type system (2.1) is encapsulated in the algebraic system (2.10) regarded as a definition of ** u**. In fact, this observation may be interpreted as a corollary of Tsarev's theorem because if we select a ‘time’

*t*

^{α0}and regard all other

*t*

^{α}s as parameters then system (2.10) may be formulated as

*α*=

*α*

_{0}.

As in the classical case (*N*=1), the algebraic system (2.10) turns out to be equivalent to a system of first-order differential equations. Indeed, if we regard (2.10) as a definition of some functions *μ*^{i} then, on substitution into the adjoint eigenfunction equations (2.2), it is readily verified that these functions constitute adjoint eigenfunctions if and only if the generalized hodograph equations

### (c) Iterated adjoint Darboux transformations

It turns out that, just like the characteristic speeds *μ*^{i}, the remaining ingredients *x* and *t*^{α} of the algebraic system (2.10) have distinct soliton-theoretic meaning. Thus, we first consider two sets {*μ*^{i}} and {λ^{i}} of adjoint eigenfunctions obeying
*A*_{ik}} of the underlying Darboux system. This system is known to be invariant under adjoint Darboux transformations [27,37]. Specifically, for fixed *i*, the adjoint Darboux transformation ^{i} transforms the adjoint eigenfunctions *μ*^{l} according to

By construction, the above Darboux transforms obey a linear system of the type (2.14) with coefficients depending on *A*_{ik} and the adjoint eigenfunctions λ^{i} only. The latter property guarantees that adjoint Darboux transformations may be iterated in the following purely algebraic manner. Given any *N* sets of eigenfunctions *N* adjoint Darboux transformations *N*th Darboux transform of the adjoint eigenfunction *μ*^{N+1} is given by
*N*th Darboux transform depends neither on the order of application of the adjoint Darboux transformations nor on the components of the sets of adjoint eigenfunctions *α*_{1},…,*α*_{N}) and (*i*_{1},…,*i*_{N+1}) of (1,…,*N*) and (1,…,*N*+1), respectively, the iterated Darboux transform
*x* and *t*^{α}. Thus, remarkably, by virtue of the commutativity of the flows (2.1), the ‘spatial’ independent variable *x* may be interpreted as the unique *N*-fold Darboux transform constructed from the characteristic speeds

The interpretation of the ‘times’ *t*^{α} is now based on the observation that the system (2.10) is implicitly symmetrical in *x* and *t*^{α}. Indeed, for any fixed *α*, the system (2.10) may be reformulated as
*x* and *t*^{α} have been interchanged. In fact, the *a priori* formal symmetry obtained in this manner may indeed be exploited by rewriting the linear system (2.3) as
*ψ*_{α}. Hence, for reasons of symmetry, the time *t*^{α} obtained by means of Cramer's rule from (2.19) or, equivalently, the original system (2.10) coincides with the iterated Darboux transform
*t*^{α} to deduce that
*N*+1 variables *x* and *t*^{α} may also be regarded as Combescure transforms of each other.

## 3. Discrete generalized hodograph equations

The formulation of the classical hodograph equations and their generalization in the language of (adjoint) eigenfunctions may instantly be used to derive their canonical integrable discrete counterparts. Indeed, the standard integrable discretization of the conjugate net equations (2.4) turns out to be the fundamental structure on which this discretization technique is based. Thus, if
_{i} are defined by Δ_{i}*ψ*=*ψ*_{[i]}−*ψ* and Δ_{ik}=Δ_{i}Δ_{k}=Δ_{k}Δ_{i}, then a discrete Combescure transform *ψ*_{⋆} of *ψ* defined by
^{i} constitute solutions of the linear system [33,39]
*i*] denotes the relative unit increment _{ik} acts according to Δ_{ik}*ψ*=*ψ*_{[ik]}−*ψ*_{[i]}−*ψ*_{[k]}+*ψ*. It is noted that (3.2) and (3.3) represent the discrete analogues of the linear equations (1.2) and (1.4) defining the densities *ψ* and fluxes *ψ*_{⋆} associated with the conservation laws for semi-Hamiltonian systems of hydrodynamic type. In connection with an appropriate Cauchy problem, it is convenient to reformulate the adjoint linear system (3.4) as
*A*_{ik} which constitutes the standard integrable discretization of the aforementioned Darboux system [33,40].

The discrete analogues of the classical adjoint Darboux transformations may be obtained by formally replacing derivatives by differences in the transformation laws (2.15). Indeed, the Darboux transforms of another set of adjoint eigenfunctions {*μ*^{l}} are given by
*i* corresponding to the adjoint eigenfunction λ^{i} which generates the adjoint Darboux transformation _{2} which coincides with the transformation law (2.15)_{2}, the expressions (2.16), (2.17) and (2.21)_{2} for the iterated Darboux transforms are also valid in the discrete case. Moreover, the quantities *x* and *t*^{α} defined by
*μ*^{i} now refer to discrete adjoint eigenfunctions. In analogy with the continuous case, insertion into the adjoint eigenfunction equations (3.5) for {*μ*^{i}} leads to the linear system
*μ*^{i}}. Finally, the discrete generalized hodograph equations adopt the form
*N*+1 variables *x* and *t*^{α} may be interpreted as discrete Combescure transforms of each other.

As pointed out in the previous section, there exists complete equivalence between the hydrodynamic-type system (2.1) and the generalized hodograph equations (2.13). In fact, this is verified directly by employing a formulation in terms of differential forms (cf. [41]). Indeed, it is seen that the system
*x* and *t*^{α} are chosen as the independent variables. Alternatively, one may select the *u*^{i}s as the independent variables, so that the generalized hodograph equations (2.13) result. Accordingly, the algebraic system (2.10) encodes the *N*+1-dimensional integral manifolds *x*,** t**)(

**) of the generalized hodograph equations (2.13) as an**

*u**N*+1-dimensional submanifold

**(**

*u**x*,

**) represents a corresponding solution of the hydrodynamic-type system (2.1). In the discrete case, the algebraic system (3.8) encapsulates ‘discrete integral manifolds’**

*t*

*u*^{Δ}=(

*δ*

^{1}

*n*

^{1},…,

*δ*

^{N+1}

*n*

^{N+1}) for prescribed lattice parameters

*δ*

^{i}. Hence, the variable

*u*^{Δ}may be regarded as a discretization of either the independent variables of the generalized hodograph equations (2.13) or the dependent variables of the system of hydrodynamic type (2.1). The latter corresponds to an ‘implicit discretization’ of the hydrodynamic-type system with variable spacing between the lattice points on the

*N*+1-dimensional submanifold

*x*and

**.**

*t*## 4. Discrete Euler–Poisson–Darboux systems

It is well known [17,18,19,20] that the characteristic speeds λ^{i} associated with the multi-phase-averaged KdV equations are related to linear hyperbolic equations of Euler–Poisson–Darboux-type. In fact, recently, it has been demonstrated [22] that these characteristics speeds may be generated by means of iterated Darboux transformations applied to separable solutions of (extended) Euler–Poisson–Darboux-type systems. It turns out that one may construct canonical discretizations of the multi-phase characteristic speeds if one carefully defines analogues of the hyperelliptic integrals associated with the underlying Riemann surfaces of genus *g*≥1. Here, we demonstrate how one may derive particular classes of discrete characteristic speeds from the discrete Euler–Poisson–Darboux-type system
*i*≠*k*∈{1,…,2*g*+1}, which include those of ‘averaged KdV’ type. Here, the constants *δ*^{i} are lattice parameters and the constants *ϵ*^{i} determine the nature of the contour integrals to be defined in §5. For *ϵ*^{i}=1/2, this leads to analogues of the above-mentioned hyperelliptic integrals. The parameters *ν*^{i} reflect the fact that it is crucial to maintain the freedom of placing the discretization points not necessarily on the vertices of the *x*^{i}=*δ*^{i}(*n*^{i}+*ν*^{i}), *y* and supplement the discrete Euler–Poisson–Darboux system by the differential–difference equations
*ϕ*=*ϕ*(** n**,

*y*) is well-defined, because the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3) remains compatible.

### (a) Separable solutions

As in the continuous case [22], we now focus on separable solutions of the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3). Thus, it is readily verified that the ansatz
*ρ* and *ρ*^{i} on *y* and *n*^{i}, respectively, leads to the first-order differential/difference equations
*ζ* being a (complex) constant of separation. The latter may be solved to obtain
_{1} reduce to
*ρ*^{i} represents a canonical discretization of (*ζ*−*x*^{i})^{−ϵi} so that

### (b) Superposition and iterated Darboux transformations

The separable solutions derived in the preceding may be superimposed to obtain large classes of solutions of the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3). Here, we consider the contour integrals
*b*_{κ} on the complex *ζ*-plane are assumed to be independent of *y* and ‘locally’ independent of ** n**, that is, we demand that

*f*. Accordingly, the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3) admits vector-valued solutions of the form

*g*−1)-fold Darboux transform [42] of any solution

*ϕ*of the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3) with respect to the independent variable

*y*is given by the compact expression

*g*−1)-fold Levy transform [27] with respect to

*y*. The Levy transforms of the particular solutions

*y*and that, by definition,

*g*=1.

The action of another Levy transformation with respect to the variable *n*^{i} now produces the *g*-fold Levy transform
*ϕ*^{0}. Here, the symbol λ^{i} has been chosen to indicate that the set {λ^{i}} will indeed be shown to constitute a set of adjoint eigenfunctions. Because the Levy transform λ^{i} may be formulated as
*α*_{1}=1, *α*_{k}=0 otherwise and *g*=1, the interpretation *H*_{2}=0 is to be adopted.

### (c) Discrete characteristic speeds

The connection with discrete characteristic speeds and the associated discrete generalized hodograph equations (3.9) is now made as follows. By construction, *x*^{i} may also be regarded as adjoint eigenfunctions of another system of conjugate net equations [37]. The analogous statement is true in the discrete case and, accordingly, the quantities λ^{i} constitute adjoint eigenfunctions associated with the discrete conjugate net equations
*A*_{ik} are related to the coefficients *B*_{ik} by
*g*+1 sets of adjoint eigenfunctions ^{i}.

In order to construct canonical sets of adjoint eigenfunctions satisfying (4.24), it is required to introduce an explicit parametrization of the functions *ρ*^{i} in the base separable solution (4.4) of the associated semi-discrete Euler–Poisson–Darboux system. Thus, in terms of Gamma functions [43], the general solution of the difference equation (4.5)_{1} formulated as
*ζ*. In fact, the multiplicative factor has been chosen in such a manner that

It has been pointed out that the Levy transforms *y*. This is due to the fact that the seed solution *ϕ*^{0} of the semi-discrete Euler–Poisson–Darboux system is a polynomial in *y* of degree at most *g*−1. A canonical way of generating an infinite number of seed solutions which admit this property is to expand the separable solution
*ζ*^{−1} is readily established by applying the asymptotic expansion (4.28) to the function *ρ*^{i} as given by (4.26) and reformulating it as an asymptotic series in *ζ*^{−1}, namely
*Γ*_{0} and *Γ*_{1} are seen to be
*Ξ*_{α}(** n**,

*y*) are polynomials in

*y*of degree

*α*if

*α*≤

*g*−1 and of degree

*g*−1 if

*α*>

*g*−1. In fact,

*Γ*

_{α,k}=0 otherwise. By construction, each coefficient

*Ξ*

_{α}constitutes a solution of the semi-discrete Euler–Poisson–Darboux system (4.1) and (4.3). For instance,

*Ξ*

_{0}=

*Γ*

_{0}=1 represents the trivial constant solution, whereas

*ϕ*

^{0}which has been used to construct the discrete characteristic speeds λ

^{i}given by (4.20). The constant

*c*

^{0}may be read off (4.32).

The general expression (4.12) for the iterated Darboux transform *ϕ*_{g−1} may be used to generate the (*g*−1)-fold Levy transform *ϕ*_{g−1}[*Ξ*_{α}] of any seed solution *Ξ*_{α}. Because the degree of *Ξ*_{α} in *y* is less than *g*, the Levy transform *ϕ*_{g−1}[*Ξ*_{α}] is independent of *y*. Hence, the procedure outlined in §4*b* may be simplified by evaluating the analogue of (4.14) at *y*=0. As a result, one is immediately led to the compact expression
*g*−1)-fold Levy transform associated with the discrete characteristic speeds λ^{i} is retrieved. We may now employ the eigenfunctions *ϕ*_{g−1}[*Ξ*_{α}] and *b* by replacing *ϕ*_{g−1}[*Ξ*_{α}] in (4.17) and (4.18). In terms of the coefficients *Γ*_{α,k} and the ratios of determinants
^{i} via
*g*+1 sets of adjoint eigenfunctions *α*=1,…,2*g*+1 gives rise to a discrete system of generalized hodograph equations (3.9) with associated ‘implicitly defined’ discrete commuting flows of hydrodynamic type. Once again, it is observed that (4.40) represents a natural discretization of the compact formulation of the corresponding characteristic speeds recorded in reference [20].

## 5. ‘Discrete’ hyperelliptic integrals and characteristic speeds of Whitham type

It has been demonstrated that the characteristic speeds *y* and, accordingly, the contour integrals
*b*_{κ} which mimic canonical cycles associated with classical hyperelliptic integrals [26]. To this end, we make the choice
*δn*^{i} in *x*^{i}=*δ*(*n*^{i}+*ν*^{i}) held constant as before. Up to the factor *ζ*^{k}, the integrand of the contour integrals (5.1) may be formulated as
*p* representing all functions *ρ*^{i} is defined by

### (a) ‘Discrete’ cycles and hyperelliptic integrals

We now introduce the ordering
*x*^{2k+1}=*δn*^{2k+1}. Hence, the separable solution (5.4) of the discrete Euler–Poisson–Darboux system (5.3) becomes
*g*=1, we obtain
*p*(*ξ*,*n*,0) is given by
*p* which make up *φ* partially cancel each other in such a manner that, as a function of *ξ*, *φ* has no zeros or poles in the region
*g* contours *b*_{κ} (on the *ξ*-plane) as closed paths of anticlockwise orientation which pass through the pairs of intervals (*n*^{2κ−1}−1,*n*^{2κ}) and *κ*=1,…,*g* as indicated in figure 1. The contour integrals

constitute ‘discrete’ analogues of the hyperelliptic integrals
*b*_{κ} essentially becoming the *b*-cycles employed in references [20,21] in the limit *n*^{2κ−1}−1,*n*^{2κ}) and *κ*=1,…,*g* correspond to the cuts (*x*^{2κ−1},*x*^{2κ}) and *g* are joined. In fact, the *ζ*-plane represents the union of the half of the upper sheet, and the half of the lower sheet which contains the *b*-cycles. This union is discontinuous between the cuts and, in the discrete case, this is reflected by the presence of poles and zeros between the intervals (*n*^{2κ−1}−1,*n*^{2κ}) and *b*-cycles are ‘locally’ independent of ** n** in the sense of (4.10), so that the ‘discrete’ hyperelliptic integrals (5.13) regarded as functions of

**are indeed solutions of the discrete Euler–Poisson–Darboux system (4.1).**

*n*The discrete hyperelliptic integrals (5.13) may be evaluated explicitly in terms of the residues of the meromorphic integrand *φ*, because one only requires the known relationships
*g*=1, the contour integral
*πi*) evaluated at the points
*K* of the first kind [43], this elliptic integral may be expressed as
^{1},λ^{2},λ^{3} by means of the continuous analogue of the Levy transformation (4.18) for *g*=1. These turn out to be the characteristic speeds in the one-phase averaged KdV equations derived by Whitham [2,8]. Thus, the discrete elliptic integral *ϕ*^{Δ} gives rise to discrete characteristic speeds of Whitham type.

It is observed that the elliptic integral *ϕ*^{O} only depends on the differences of the *x*^{i}. In fact, the same applies, *mutatis mutandis*, to the discrete elliptic integral (5.16), because the function *p*(*ξ*,*n*,*ν*) is invariant under a shift of *ξ* and *n* by the same amount. Accordingly, it is natural to regard the (discrete) elliptic integrals *ϕ*^{Δ} and *ϕ*^{O} as functions of the differences *n*^{2}−*n*^{1} and *n*^{3}−*n*^{2}. Their graphs are displayed in figure 2 and it its seen that there exists virtually no difference between the discrete and continuous elliptic integrals represented by points and a mesh, respectively. It is noted that this statement is independent of the lattice parameter *δ* in the sense that, as a function of ** n**, the ratio

*ϕ*

^{Δ}/

*ϕ*

^{O}does not depend on

*δ*. Thus, remarkably, there exists a unique relationship between the discrete and continuous elliptic integrals. We conclude with the remark that the summation involved in the determination of the discrete hyperelliptic integrals may be reformulated, so that it becomes transparent that the discrete hyperelliptic integrals may be expressed in terms of generalized hypergeometric functions [43].

### (b) Even number of branch points

It is natural to inquire as to the existence of contour integrals of the type (5.13) which may be regarded as the analogues of hyperelliptic integrals associated with an even number 2*g*+2 of branch points. These hyperelliptic integrals arise in connection with the multi-phase-averaged NLS equations [46]. In principle, the analogue of the solution (5.8) of the discrete Euler–Poisson–Darboux system, that is,
*ξ* which has zeros and poles at half-integers and integers, respectively. By virtue of the symmetries of the Gamma function, it turns out natural to introduce the ‘complementary’ function
*p*(*ξ*,*n*,*ν*) by
*n*^{2κ−1}−1,*n*^{2κ}). It is noted that, for convenience, the scaling of *q* has been chosen in such a manner that it approximates the function (*x*−*ζ*)^{−1/2} rather than (*ζ*−*x*)^{−1/2} in the sense of (4.27). Furthermore, up to a sign, *φ* is symmetric in *p* and *q* due to the identity
*m* and *n*. Once again, in the simplest case *g*=1, the contour integral (5.16), where the contour *b*_{1} passes anticlockwise through the intervals (*n*^{1}−1,*n*^{2}) and (*n*^{3}−1,*n*^{4}), turns out to be a very good approximation of the corresponding elliptic integral
*b*-cycles are defined as closed paths of anticlockwise orientation passing through the pairs of intervals (*n*^{2κ−1}−1,*n*^{2κ}) and (*n*^{2g+1}−1,*n*^{2g+2}) for *κ*=1,…,*g*.

## 6. Perspectives

We conclude with a selection of open problems which naturally arise in connection with the theory presented in this paper. For instance, it has been pointed out in references [2,24,22,47] that the theory of semi-Hamiltonian systems of hydrodynamic type is closely related to the analysis of the critical points of appropriately chosen functions. In the current context, if *ψ* is an eigenfunction satisfying the discrete conjugate net equations (4.22) and {*μ*^{i}}, *ψ*_{α} and *Θ* is now defined by
*a priori*, *x* and *t*^{α} are merely parameters. In analogy with the continuous case, critical points *n*_{c} of the function *Θ* are defined as points ** n**, where the ‘discrete derivatives’ of

*Θ*vanish, that is, Δ

_{i}

*Θ*|

_{n=nc}=0. Accordingly, we obtain

*x*and

*t*

^{α}to

*n*_{c}in the same manner (with the index on

*n*_{c}being dropped) as the algebraic system (3.8) which gives rise to the discrete generalized hodograph equations (3.9). The implications of this observation are currently being investigated.

In the preceding, we have regarded ‘complete’ hyperelliptic integrals as functions of their branch points *x*^{i} and, in this context, put forward a canonical definition of their discrete analogues. It is natural to inquire as to the existence of similar analogues of ‘incomplete’ hyperelliptic integrals and their associated differential equations. For instance, in the classical case, elliptic integrals are related by inversion to the differential equation

In §5, we have confined ourselves to a detailed discussion of the relevance of the discrete Euler–Poisson–Darboux-type system (4.1) for *ϵ*^{i}=1/*M*, where *M* is a positive integer, are obtained in terms of the superelliptic (*M*,*N*)-curves
*M*=2, the corresponding superelliptic integrals are relevant in the theory of Whitham-type equations. For instance, trigonal curves (3,*N*) appear in connection with the Benney equations and the dispersionless Boussinesq hierarchy (see, e.g. [48,49] and references therein). It is therefore desirable to investigate whether it is possible to extend the theory developed in this paper to define canonical discrete analogues of superelliptic integrals and associated discrete characteristic speeds of Whitham type.

## Acknowledgements

B.G.K. acknowledges support by the PRIN 2010/2011 grant no. 2010JJ4KBA_003. W.K.S. expresses his gratitude to the DFG Collaborative Research Centre SFB/TRR 109 *Discretization in Geometry and Dynamics* for its support and hospitality.

- Received July 3, 2014.
- Accepted September 15, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.